Question

Two negative integers are five minutes apart on the number line, and their product is 126. What is the sum of the two integers?

Answer 1

First I will assume that this problem can be reworded as follows:  "Two negative integers are five units apart on the number line."

Let the first (the smaller) negative integer be x and the second y.  Note that y, being bigger than x, can be represented by y=x+5.

First find the 2 integers.  Their product is 126, so x(x+5)=126.
Expanding, x^2 + 5x - 126 = 0.  Using the quadratic formula to solve this, 

          -5 plus or minus sqrt(529)        -5 plus or minus 23
x = --------------------------------------- = -----------------------------
                           2                                              2
                                                                                          -5 -23
We are interested only in negative integers, so take x = ------------
                                                                                               2

Then x = -28/2, or x = -14.  Then y is x+5, or -14+5 = -9.

Check:  does the product of -9 and -14 equal 126, as it must?   YES.